PERIODIC MOTION 329 mass. Find the period with which the particle will vibrate when displaced along the string. Let L' be the stretched length of the string, A the area of its cross-section, q the distance of the particle from its position of equilibrium, and T\ and jP2 the tensile forces of the two parts of the string. Then by Hooke's law we have £' \ L 1 y I n T t 7" ! O ~ T = X -----r-A L 2 2 __ , L' -L~-2q \ 2 Therefore the resultant force on the particle is T: A = -i^ -- ITg==~~L" where X' = ^LX. Hence the potential energy equals But since the kinetic energy is given by 1 2X' we obtain E = - mq* + — q2 Al JLJ for the total energy of the system. Differentiating the last equation we get mg + ^-o which gives ___ q^asm^—tt + to) and P = 3. A cylinder performs small oscillations inside of a fixed cylinder. Find the period of the motion, supposing the contact between the cylinders to be rough enough to prevent sliding.